Must clothing be washed if it becomes soiled by the bottom of someone else's shoe, or is it sufficient to wipe the soiled area or sprinkle it with water?

The default state of footwear and the ground is purity, and impurity is not to be declared except with certainty. Therefore, anything that touches clothes from people's shoes is considered pure and does not require washing or investigating its purity. It is sufficient to act upon the default, which is purity. Ibn al-Qayyim mentioned the story of Umar ibn al-Khattab, may Allah be pleased with him, when something fell on him from a spout, and he said, "O owner of the spout, do not tell us," and he walked away. This indicates that there is no obligation to investigate or inquire about the purity of something that befalls a person if its true nature is unknown. Thus, one should dismiss the thought in this matter, and there is no harm in wiping off what touches you out of precaution, not obligation.